Machine Kinematics
solved, then drawn
Concept

Gough platform — where it appears

A platform carried by six legs of adjustable length, each with a spherical joint at both ends. Its inverse problem is six independent square roots and its forward problem has up to forty solutions, which is the parallel field's standing example of the two problems changing places.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

What became of Bézout's paths. four-bar coupler pin: 2 of 4 paths arrived at a solution and 2 went to infinity; 3-RPR platform: 6 of 16 paths arrived at a solution and 10 went to infinity; Gough, generic: 80 of 1458 paths arrived at a solution and 1378 went to infinity. The surplus is not merely wasted — it is cheap: a path on its way to infinity is abandoned in a handful of steps, while every path that arrives is tracked in full.

The paths that leave

Bézout's number over-counts, and the over-count is enormous — 1,458 tracked paths for 80 solutions. The obvious response is to find a method that tracks only the paths that arrive. That method exists, it was built, and it is four times slower, because the surplus paths are not merely surplus. They are cheap.

algebra · Algebra The twelve that were at infinity, coming back. Every solution of the platform's direct kinematics, plotted by how far from the origin it sits, as the six anchors are jittered. At no jitter the site's own platform has 56 solutions and the largest is at 25.8. At a jitter of 0.2 there are 80, and the extra ones arrive from far out — they were never missing, they were at infinity.

Twenty-eight, not forty

The general six-legged platform has forty poses for a given set of leg lengths, and this site has quoted that number beside a picture of a platform that has twenty-eight. Its anchors are arranged symmetrically, which makes it a special architecture, and the missing twelve poses are not missing. They are at infinity, and perturbing the anchors brings them back.

algebra · Algebra A lower bound that happened to be tight. The site's own Gough platform at its home pose. The search starts Newton from a spread of guesses and reports what it lands on: 16 from 400, 16 from 1200, 16 from 4000. Tracking every one of the 1458 Bézout paths says there are 28 poses in the complex numbers and 16 of them are real. The search was right. Nothing available to the search could have said so.

The search that was right

This site has reported sixteen assemblies for its Gough platform and labelled the number a lower bound found by search, everywhere it appears. Tracking every path says there are twenty-eight poses and sixteen of them are real. The lower bound was tight. Nothing available to the search could have said so, and a second method that was supposed to settle it turns out to have the same defect one level up.

algebra · Algebra

Named alongside it

The objects these essays reach for when they reach for this one.

BezoutDirect kinematicsHomotopy continuationMonodromyPolynomial systemSolutions at infinityAssembly branchComplex solutionComputational costLower boundNewton–RaphsonPath tracking

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